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<article class="li"><h3 class="heading">
<span class="type">Item</span><span class="period">.</span>
</h3>
<p>The <dfn class="terminology">radius of convergence</dfn> (about <span class="process-math">\(x_0\)</span>): a nonnegative <span class="process-math">\(\textcolor{blue}{\rho}\text{,}\)</span> such that</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\sum_{n=0}^{\infty} a_n(x-x_0)^n ~~\left\{\begin{array}{ll} \text{\textcolor{blue}{converges absolutely}} &amp; \text{for $\textcolor{magenta}{|x-x_0|&lt;}\textcolor{blue}{\rho}$},\\ \text{diverges} &amp;\text{for $|x-x_0|&gt;\textcolor{blue}{\rho}$.}\end{array}\right.
\end{equation*}
</div>
<p>For a series that converges only at <span class="process-math">\(x_0\text{,}\)</span> <span class="process-math">\(\rho=0\text{;}\)</span>for a series that converges for all <span class="process-math">\(x\text{,}\)</span> <span class="process-math">\(\rho\)</span> is infinite.</p>
<p>If <span class="process-math">\(\rho&gt;0\text{,}\)</span> then the interval <span class="process-math">\(|x-x_0|&lt;\rho\)</span> is called <dfn class="terminology">the interval of convergence</dfn>.</p></article><span class="incontext"><a href="sec5_1.html#li-7" class="internal">in-context</a></span>
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